Inequalities of Hermite-Hadamard Type for GG-Convex Functions

Autor: Sever S Dragomir
Rok vydání: 2019
Předmět:
Zdroj: Annals of the West University of Timisoara: Mathematics and Computer Science, Vol 57, Iss 2, Pp 34-52 (2019)
ISSN: 1841-3307
DOI: 10.2478/awutm-2019-0013
Popis: Some inequalities of Hermite-Hadamard type for GA-convex functions de…ned on positive intervals are given. 1. Introduction Let I (0;1) be an interval; a real-valued function f : I ! R is said to be GA-convex (concave) on I if (1.1) f x y ( ) (1 ) f (x) + f (y) for all x; y 2 I and 2 [0; 1]. Since the condition (1.1) can be written as (1.2) f exp ((1 ) lnx+ ln y) ( ) (1 ) f exp (lnx) + f exp (ln y) ; then we observe that f : I ! R is GA-convex (concave) on I if and only if f exp is convex (concave) on ln I := fln z; z 2 Ig : If I = [a; b] then ln I = [ln a; ln b] : It is known that the function f (x) = ln (1 + x) is GA-convex on (0;1) [4]. For real and positive values of x, the Euler gamma function and its logarithmic derivative , the so-called digamma function, are de…ned by (x) := Z 1 0 t e dt and (x) := 0 (x) (x) : It has been shown in [54] that the function f : (0;1)! R de…ned by f (x) = (x) + 1 2x is GA-concave on (0;1) while the function g : (0;1)! R de…ned by g (x) = (x) + 1 2x + 1 12x2 is GA-convex on (0;1) : If [a; b] (0;1) and the function g : [ln a; ln b] ! R is convex (concave) on [ln a; ln b] ; then the function f : [a; b] ! R, f (t) = g (ln t) is GA-convex (concave) on [a; b] : Indeed, if x; y 2 [a; b] and 2 [0; 1] ; then f x y = g ln x y = g [(1 ) lnx+ ln y] ( ) (1 ) g (lnx) + g (ln y) = (1 ) f (x) + f (y) showing that f is GA-convex (concave) on [a; b] : 1991 Mathematics Subject Classi…cation. 26D15; 25D10. Key words and phrases. Convex functions, Integral inequalities, GA-Convex functions. 1 2 S. S. DRAGOMIR We recall that the classical Hermite-Hadamard inequality that states that (1.3) f a+ b 2 1 b a Z b a f (t) dt f (a) + f (b) 2 for any convex function f : [a; b]! R. For related results, see [1]-[20], [23]-[25], [26]-[35] and [36]-[46]. In [54] the authors obtained the following Hermite-Hadamard type inequality. Theorem 1. If b > a > 0 and f : [a; b]! R is a di¤erentiable GA-convex (concave) function on [a; b] ; then (1.4) f (I (a; b)) ( ) 1 b a Z b a f (t) dt ( ) b L (a; b) b a f (b)+ L (a; b) a b a f (a) : The identric mean I (a; b) is de…ned by I (a; b) := 1 e b aa 1 b a while the logarithmic mean is de…ned by L (a; b) := b a ln b ln a The di¤erentiability of the function is not necessary in Theorem 1 for the …rst inequality (1.4) to hold. A proof of this fact is proved below after some short preliminaries. The second inequality in (1.4) has been proved in [54] without differentiability assumption. 2. Preliminaries We recall some facts on the lateral derivatives of a convex function. Suppose that I is an interval of real numbers with interior I and f : I ! R is a convex function on I. Then f is continuous on I and has …nite left and right derivatives at each point of I. Moreover, if x; y 2 I and x < y; then f 0 (x) f 0 + (x) f 0 (y) f 0 + (y) which shows that both f 0 and f 0 + are nondecreasing function on I. It is also known that a convex function must be di¤erentiable except for at most countably many points. For a convex function f : I ! R, the subdi¤erential of f denoted by @f is the set of all functions ' : I ! [ 1;1] such that ' °I R and f (x) f (a) + (x a)' (a) for any x; a 2 I: It is also well known that if f is convex on I; then @f is nonempty, f 0 , f 0 + 2 @f and if ' 2 @f , then f 0 (x) ' (x) f 0 + (x) for any x 2 I. In particular, ' is a nondecreasing function. If f is di¤erentiable and convex on I, then @f = ff 0g : Now, since f exp is convex on [ln a; ln b] it follows that f has …nite lateral derivatives on (ln a; ln b) and by gradient inequality for convex functions we have (2.1) f exp (x) f exp (y) (x y)' (exp y) exp y where ' (exp y) 2 f 0 (exp y) ; f 0 + (exp y) for any x; y 2 (ln a; ln b) : INEQUALITIES OF HERMITE-HADAMARD TYPE FOR GA-CONVEX FUNCTIONS 3 If s; t 2 (a; b) and we take in (2.1) x = ln t; y = ln s; then we get (2.2) f (t) f (s) (ln t ln s)' (s) s where ' (s) 2 f 0 (s) ; f 0 + (s) : Now, if we take the integral mean on [a; b] in the inequality (2.2) we get 1 b a Z b a f (t) dt f (s) 1 b a Z b a ln tdt ln s ! ' (s) s and since 1 b a Z b a ln tdt = ln I (a; b) then we get (2.3) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s)' (s) s for any s 2 (a; b) and ' (s) 2 f 0 (s) ; f 0 + (s) : This is an inequality of interest in itself. Now, if we take in (2.3) s = I (a; b) 2 (a; b) then we get the …rst inequality in (1.4) for GA-convex functions. If f is di¤erentiable and GA-convex on (a; b) ; then we have from (2.3) the inequality (2.4) 1 b a Z b a f (t) dt f (s) + (ln I (a; b) ln s) f 0 (s) s for any s 2 (a; b) : If we take in (2.4) s = a+b 2 = A (a; b) ; then we get (2.5) 1 b a Z b a f (t) dt f (A (a; b)) f 0 (A (a; b))A (a; b) ln A (a; b) I (a; b) : If we assume that f 0 (A (a; b)) 0; then, since I (a; b) A (a; b) ; we get (2.6) 1 b a Z b a f (t) dt f (A (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Also, if we take in (2.4) s = L (a; b) ; then we get (2.7) 1 b a Z b a f (t) dt f (L (a; b)) + f 0 (L (a; b))L (a; b) ln I (a; b) L (a; b) : If we assume that f 0 (L (a; b)) 0; then we get from (2.7) that (2.8) 1 b a Z b a f (t) dt f (L (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Now, if we take in (2.4) s = p ab = G (a; b) ; then we get (2.9) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) ln I (a; b) G (a; b) : 4 S. S. DRAGOMIR Since ln I (a; b) G (a; b) = ln I (a; b) lnG (a; b) = b ln b a ln a b a 1 ln a+ ln b 2 = a+ b 2 ln b ln a b a 1 = A (a; b) L (a; b) L (a; b) ; then (2.9) is equivalent to (2.10) 1 b a Z b a f (t) dt f (G (a; b)) + f 0 (G (a; b))G (a; b) A (a; b) L (a; b) L (a; b) : If f 0 (G (a; b)) 0; then we have (2.11) 1 b a Z b a f (t) dt f (G (a; b)) provided that f is di¤erentiable and GA-convex on (a; b) : Motivated by the above results we establish in this paper other inequalities of Hermite-Hadamard type for GA-convex functions. Applications for special means are also provided. 3. New Results We start with the following result that provide in the right side of (1.4) a bound in terms of the identric mean. Theorem 2. Let f : [a; b] (0;1) ! R be a GA-convex (concave) function on [a; b] : Then we have 1 b a Z b a f (t) dt ( ) (ln b ln I (a; b)) f (a) + (ln I (a; b) ln a) f (b) ln b ln a (3.1) = b L (a; b) b a f (b) + L (a; b) a b a f (a) : Proof. Since is a GA-convex (concave) function on [a; b] then f exp is convex (concave) and we have f (t) = f exp (ln t) = f exp (ln b ln t) ln a+ (ln t ln a) ln b ln b ln a (3.2) ( ) (ln b ln t) f exp (ln a) + (ln t ln a) f exp (ln b) ln b ln a = (ln b ln t) f (a) + (ln t ln a) f (b) ln b ln a for any t 2 [a; b] : This inequality is of interest in itself as well. If we take the integral mean in (3.2) we get 1 b a Z b
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